Joint probability distributions
Def: the function f(x,y) is a joint probability distribution or p.m.f of the discrete r.v. X and Y if
f(x,y)?0 for all (x,y)
?_x??_y??f(x,y)=1?
P(X=x,Y=y)=f(x,y)
For any region A in the xy plan, p[(x,y)?A]= ???_A?f(x,y) .
Example:-suppose
P(x1,x2)=(x1+x2)/21,x1=1,x2=2 is j.p.m.f for two randomes x1,x2
P(X1=x1,X2=x2)=P(X1=1,X2=2)=1/7, P(X1=x1,X2=x2)>0.
x1=1,3 x2=1,2
?_(x1=1)??_(x2=1)??f(x,y)=?_(x1=1)^3??_(x2=1)^2??(x1+x2)/21=1/21 ?_(x1=1)^3??(2x1+3)=1/21.21=1???
(x1,x2) = (1,1) (1,2) (2,1) (2,2) (3,1) (3,2)
P(x1,x2)= 2/21 3/21 3/21 4/21 4/21 5/21.
Def:- the function f(x,y) is a joint density function of the continuous random variables X and Y if
f(x,y)?0 for all (x,y)
?_(-?)^???_(-?)^???f(x,y)dx dy?=1
P(X,Y)?A=?_(-?)^???_(-?)^???f(x,y)dx dy?
Joint probability distributions
Def: the function f(x,y) is a joint probability distribution or p.m.f of the discrete r.v. X and Y if
f(x,y)?0 for all (x,y)
?_x??_y??f(x,y)=1?
P(X=x,Y=y)=f(x,y)
For any region A in the xy plan, p[(x,y)?A]= ???_A?f(x,y) .
Example:-suppose
P(x1,x2)=(x1+x2)/21,x1=1,x2=2 is j.p.m.f for two randomes x1,x2
P(X1=x1,X2=x2)=P(X1=1,X2=2)=1/7, P(X1=x1,X2=x2)>0.
x1=1,3 x2=1,2
?_(x1=1)??_(x2=1)??f(x,y)=?_(x1=1)^3??_(x2=1)^2??(x1+x2)/21=1/21 ?_(x1=1)^3??(2x1+3)=1/21.21=1???
(x1,x2) = (1,1) (1,2) (2,1) (2,2) (3,1) (3,2)
P(x1,x2)= 2/21 3/21 3/21 4/21 4/21 5/21.
Def:- the function f(x,y) is a joint density function of the continuous random variables X and Y if
f(x,y)?0 for all (x,y)
?_(-?)^???_(-?)^???f(x,y)dx dy?=1
P(X,Y)?A=?_(-?)^???_(-?)^???f(x,y)dx dy?